$A$ round table conference is to be held amongst $20$ countries. If two particular delegates wish to sit together,then in how many ways can such arrangements be done?

In how many ways can $5$ keys be arranged in a ring?

The number of ways in which $6$ boys and $4$ girls can be seated around a round table such that $2$ special boys and a special girl never sit together is

If $3$ sisters and $8$ brothers are together playing a game,then the number of ways in which all the sisters and brothers are to be seated around a circle such that all the $3$ sisters are not seated together is

$A$ family consisting of a mother,father,and their $8$ children ($4$ boys and $4$ girls) are to be seated at a round table in a party. How many ways can this be done if the mother and father sit together and the males and females alternate?

If $3$ sisters and $8$ other girls are together playing a game,then the number of ways in which all the girls are seated around a circle such that the three sisters are not seated together,is